Absoluteness, Potential Scott Sentences, and Stability in Model Theory

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The proposed research is in model theory, which is a branch of mathematical logic. Much of model theory concerns the ways in which a theory, which is simply a set of sentences in a formal language controls its class of models. For many years, the PI has concentrated on mechanisms by which a theory can either admit or forbid certain combinatorial configurations in its models, and the proposed research is to continue these investigations in several disparate contexts. In some cases, this investigation melds well with computational learning theory. As one example, if a theory forbids the independence property, then all of the concepts i.e., definable sets, arising in the models of the theory are PAC learnable.In more detail, potential canonical Scott sentences have proved to be a useful tool in determining the Borel complexity of invariant classes of countable structures and we intend to streamline these methods by exploring thickness and groundedness of classes of models. It is hoped that the Borel complexity of every mutually algebraic theory can be computed. Theories with non-maximal uncountable spectrum are classifiable. Recent technical results about the existence of prime models make it tractable to settle Vaught's conjecture for classifiable theories and possibly for superstable theories as well. In first order logic, aleph1-categoricity of a theory is an absolute notion, as can be seen by the Baldwin-Lachlan characterization of aleph1-categoricity. The PI proposes to determine whether a similar characterization can be found for aleph1-categoricity of sentences of L(omega1, omega), or equivalently for classes of atomic models. Specifically, the proposed research is to determine whether aleph1-categoricity is absolute for classes of atomic models. The PI also proposes to use previous results about mutually algebraic structures to study expansions of weakly minimal structures that preserve stability.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.